$f(x) = \begin{cases} -8 & \text{if } x = 1 \\ -3x^{2}+1 & \text{otherwise} \end{cases}$ What is the range of $f(x)$ ?
Solution: First consider the behavior for $x \ne 1$ Consider the range of $-3x^{2}$ The range of $x^2$ is $\{\, y \mid y \ge 0 \,\}$ Multiplying by $-3$ flips the range to $\{\, y \mid y \le 0 \,\}$ To get $-3x^{2}+1$ , we add $1$ If $x = 1$, then $f(x) = -8$. Since $-8 ≤ 1$, the range is still $\{\, y \mid y ≤ 1 \,\}$.